Novel synchronous CDMA multiuser detection scheme: orthogonal decision-feedback detection and its performance study

X.H.Chen H.K.Sim

Indexing terms: CDMA, Multiuser detection, Decision feedback, Signal detection

Abstract: The authors propose a novel multiuser detection scheme for synchronous DS/CDMA systems, the orthogonal decision-feedback detector (ODFD), which performs equally well as the decorrelating decision-feedback detector (DDFD) but with a much reduced complexity. The ODFD employs a match filter bank matching to a set of orthonormal sequences which are generated by the Gram-Schmidt orthogonalisation procedure based on spreading codes. The ODFD algorithm involves only the orthonormal coefficient matrix which requires no frequent recalculations even when system parameters change. Successive decision-feedback detection is performed immediately at the output of the ODFD match filter bank without the transform matrix which is required in the DDFD.

1 Introduction

The conventional correlator receivers in a CDMA sys- tem suffer from the ‘near-far’ problem and thus require strict power control to ensure an acceptable perform- ance. Multiuser detection has been proposed for the CDMA systems [l-51 to mitigate the ‘near-far’ prob- lem. However, the concern on a CDMA multiuser detector is its algorithm complexity. The optimum receiver implemented by the multiuser Viterbi algo- rithm is prohibitively complex [l]. A linear suboptimal solution, the decorrelating detector (DD), was pro- posed [2] which is simpler and outperforms the conven- tional correlator receiver with its near-optimal ‘near- far’ resistance. However, its algorithm enhances noise due to channel inversion operation. Reported nonlinear suboptimal multiuser detectors include multistage detectors [ 3 , 41, successive interference cancellers [5] and decorrelating decision-feedback detectors (DDFD) [6, 71, all of which utilise feedback channel(s) to cancel multiple access interference (MAI) and offer a perfonn- 0 IEE, 1997 IEE Proceedings online no. 1997 1247 Paper first received 2nd April and in revised form 10th December 1996 X.H. Chen was with the National University of Singapore and is now with the Department of Electrical Engineering, National Chug-Hsing University, Tai-Chung, Taiwan, Republic of China H.K. Sim is with the Department of Electrical Engineering, National Uni- versity of Singapore, Singapore 119260

ance close to the optimum detector. The multistage detector is more complex than the DDFD. Although simpler in hardware than the DDFD, a serial succes- sive interference canceller probably introduces an intol- erable delay. In this paper, we will compare our new scheme, the orthogonal decision-feedback detector (ODFD), with the DDFD.

The DDFD [6] consists of two units. The first is the decorrelating unit and the second is the decision-feed- back unit. The DDFD detects signals in order of decreasing received signal power. Unfortunately, the DDFD algorithm needs a transform matrix with a complexity growing exponentially with the number of users. Moreover, whenever the number and power level of the users in the system change, the transform matrix has to be recalculated and it consumes a lot of time. Therefore, we are motivated to simplify the detector by eliminating the transform matrix in the DDFD.

The proposed ODFD eliminates the transform matrix by using an orthonormal match filter bank. Dif- ferent from the DDFD where its match filter bank matches the signature sequences of the users, the ODFD employs a match filter bank matching to a set of orthonormal sequences which span the signal space for the set of signature sequences. The users in the ODFD are sorted in an increasing power order and the last user (the strongest) is detected first. The results show that the ODFD has a similar performance to that of the DDFD, but with a much simpler structure (with- out the transform matrix), due to the use of the orthonormal match filter bank. An improved DDFD (IDDFD) [7] can alleviate some problems existing in the DDFD, such as power order sensitive performance and uneven detection efficiency among users. The ODFD can also be improved further by taking the same measures used in the IDDFD.

2 ODFD system model

Assume that there are K users in a synchronous CDMA system and that each uses a preassigned nor- malised signature waveform sk(t) (k = 1, 2, ..., K ) as its spreading code. Each waveform is restricted to a sym- bol interval of duration T, so that

JUT &t)d t = 1 (1)

The antipodal binary signals are considered such that binary bits 1 and 0 are represented by +1 and -1, respectively. If the input bit vector is b = (bl , b2, ..., bk,

215 IEE Proc.-Commun., Vol. 144, No. 4, August 1997

..., bK)T where bk = {-1, + I } is from the kth user, the baseband signal of the kth user in a period of Tis

where E,' is the received power and thus dEk is the channel attenuation factor of the kth user. It is assumed that a receiver knows the power of the received signals. The received baseband signal is the sum of all the user signals plus the additive white Gaussian noise (AWGN),

~ k ( t ) = b k f i S k ( t ) ( k = 1,2,. . . , K ) (2)

K

( 3 ) Ic=l

In ODFD, the independent signature waveforms, sk(t) (k = 1, 2, ..., K) , are represented by K orthonormal functions #l(t) ( i = 1, 2, ..., K ) which are defined on [0, f l and satisfy:

Therefore, the K orthonormal functions #l(t) (i = 1, 2, ..., K ) span a K-dimensional signal space for the signa- ture waveforms sk(t) (k = l , 2, ..., K ) , each of which, say sk(t), can be expressed by the orthonormal func- tions so that

K

4) = S n , k d n ( t ) ( k = L2,. . . , K ) (5) n=l

where T

sm,k = 1 S k ( t ) $ m ( t ) d t m = 172,. . . , K ) (6)

By using the orthonormal functions, eqn. 3 can be rewritten as

K K

k=l n=l

K r~ 1

n=l Lk=1 J If the received signal passes through K matched filters matching to the set of the K orthonormal functions, the output from the ith filter is

K

s , ,kbk& + n, ( i = 1,2,. . . , K ) (8) k = l

where T

n, = 1 n(t)d,(t)dt (i = 1 , 2 , . . . , K ) (9) is a Gaussian random variable. The output vector from this orthonormal matched filter bank is

. . . bK ,:I + . . .

n K zt 1 (11)

Here the vector n is a Gaussian vector with autocorre- lation matrix R(n) = a21 where I is a K x K identity

216

matrix. Similar to the DD [Z], the input data vector b can be decoded by multiplying r with the matrix filter S-l, followed by a set of decision devices. However, the complexity due to such linear detection method will just be the same as that of the DD. Therefore, it is not in our best interest to do so. Since the Gram-Schmidt orthogonalisation procedure [8] is applied to determine the matrix S, which is already an upper triangle matrix with zeros at the lower left triangle, a much simpler algorithm can be used in the ODFD than multiplying r with the inverse matrix S-l.

bK Fig. 1 Block diagram of orthogonal decision feedback detector (ODFD)

Successive decision feedback can be applied to the ODFD scheme. The detection starts with the last user (the strongest user) without using the transform matrix which is needed in the DDFD. The users are sorted in an increasing power order. Under the ODFD algo- rithm whose block diagram is shown in Fig. 1, the kth component of r is:

n= k which is different from eqn. 7 since the number of terms in eqn. 12 is reduced. The last or Kth component of r is just

T K = s K , K ~ K ~ E K + n K (13) which does not involve with the other users, so that its decision can be made right away. Since the channel attenuation factor dEK and sK,K are positive (the Gram-Schmidt procedure [8] guarantees to generate all positive diagonal terms), the decision is simply

which can be subsequently fed back to detect the ( K - 1)th user signal as follows:

(15) In general, the kth user will use the previous decisions to obtain its own estimate as:

6 K = s g n ( r K ) (14)

~ K - I = sgn ( TK-i - S K - I , K & ~ K )

Therefore, similarly to the DDFD, the ODFD algo- rithm also needs a correct power ordering, and the first detected (the strongest) user does not benefit from the decision feedback. An improved DDFD scheme, IDDFD, was proposed in [7] to overcome the prob- lems. The improved measure used in the IDDFD is also applicable to ODFD.

The Euclidean metric of the ODFD algorithm is ~r K

k=l L n=k The original idea for the decision feedback in both ODFD and DDFD is to perform a limited tree search 4

IEE Proc - ~ 6 r " n . , Vol 144, No 4, August 1997

by just keeping one path. This means that the value of bk (either +1 or -1) is selected to minimise the metric:

r K

L n=k+l

where {i&+l, ik+2 , ..., iK} are the previously detected symbols such that the Euclidean metric sum, (ml + m2 +...+ mK), is minimised. In the improved ODFD or DDFD scheme, instead of just keeping one path in the limited tree search, more paths are utilised. It has been shown [7] that using just two paths (NF = 2) is already enough to yield a satisfactory performance. Thus, we will only consider the number of the paths (NF) up to 2 in our analysis.

3 Performance and complexity

In this paper, simulation is used to compare the ODFD with the DDFD. To reveal how efficiently the detection schemes eliminate the MAI, the single user bound is taken as the lower BER bound, which is given by

I -\

= Q (J2) The flowchart of the simulation program is shown in Fig. 2, which is applicable to both the (improved) ODFD and DDFD. Various spreading codes, such as Gold code, Kasami code, M-sequence, GMW code and NO code, are considered in the simulation. The degree of these codes is either 5 or 6. To make an unbiased comparison, the power ordering of the users in the ODFD is just the opposite to that in the DDFD, because they use reverse decoding sequences. Thus, the first user in the DDFD is equivalent to the last user in the ODFD.

r calculate received vector

find metric sums select NF vectors

I 1 6 seled decoded vector

Fig. 2 Simulation flowchart for ODFD/DDFD detection

In the ODFD (or DDFD) scheme, the matrix S (or F in DDFD) is determined using the Gram-Schmidt pro- cedure (or the Cholesky algorithm in DDFD). Then the energy matrix E is calculated by

The AWGN variance d is equal to the unit. In this

IEE Proc.-Commun., Vol. 144, No. 4, August 1997

paper, we want to look into the two cases addressed by [7]. The first case is that the MA1 level is comparable to the signal to be decoded (equal power case) and the second is that the ordering is incorrect (unequal power case). In the unequal power case, the power of the last (or first in DDFD) user is 3dB less than the power of the rest.

The elements in the input data vector b are generated randomly and take values from the set {+l, -1} with equal probability. The elements in the AWGN vector n obey the normal distribution with zero-mean and unit variance. The received vector r can then be calculated by SEb + n (or FEb + n in DDFD), as shown in eqn. 10 and eqn. 11. With this received vector r we can pro- ceed to decode the signal.

The metric mK (or ml' in DDFD) is calculated first using eqn. 18 (or eqn. 7 in [7] for DDFD) for bK (or bl in DDFD). At the first stage, the metric sum is always equal to the metric itself, since only one metric term exists. Then, we select NF estimates of bK (or bl in DDFD) which correspond to the NF smallest values of the metric sums and feed them into theAnext- stage. From the obtained 2NF possible vectors [bK-l, bK] (or [b l , b2] in DDFD) the metric sums mK-l + mK (or ml + m2' in DDFD) are calculated. Again, NF smallest val- ues of the metric sums a_re selectcd to form their corre- sponding vectors [bK-l, bK] (or [bl, b2] in DDFD) and fed to the next stage. This procedure repeats until the first (or the last in DDFD) user is reached and, at this point, only one vector is selected as the recovered data vector. The decoded bit is compared with the original bit sent. If they do not match, the number of errors is incremented. The simulation ends with either 500 errors or 2 000 000 trials accumulated, whichever is reached first.

It is noted that the orthonormal matrix S in ODFD is similar to the filter matrix F in the DDFD. However, S retains many advantages over F as S needs no com- plete update even when the number of users changes, whereas the matrix F does. For example, for a two-user system, the S matrix in the ODFD is

When a new (or the third) user enters the system, the two existing orthonormal functions remain unchanged. We only need to determine the third orthonormal func- tion based on the third signature sequence. Therefore, only the last column of the S matrix needs to be deter- mined such that

s1,l s1 ,2 s1,3

s = [ ; 5.;. (22)

Hence, the values of q1, s1,2 and s2,2 (used in the feed- back previously) need not be recalculated. Therefore, if applying the decorrelating method to the ODFD, the filter matrix, (ST)-', does not need to be recalculated completely and the matrix changes simply from

to 0

(ST)-' = [ $ s ; , ~ ] (24)

where we need only to determine ~ ' 3 , 1 , ~ ' 3 , 2 and s ' ~ , ~ . As a result, the algorithm complexity becomes truly linear

4 , 2 4 , 3

217

to the number of users, since the number of terms to be determined is exactly equal to the number of users.

On the other hand, when users leave the system, we can either recalculate the matrix or simply proceed with the existing matrix. For example, when the second user in the three-user system concerned above leaves, eqn. 7

-8 (VI) Le': I , I I I T

V VU" I

0 00001

O 1 I'l; + 1:;l n1 1 n2 J

E L 1 1 b 3 1 1.31

n3

(25) Using the last element of r, b3 can be estimated and fed back to find bl. Note that the third element (s3,3E3,363) was used in the process, instead of the second element ( s ~ , ~ E ~ , ~ ~ ~ ) , because s3,3 is always positive and larger than ~2,3, which may even be negative. However, it should be admitted that the detection can be made eas- ier if the matrix is recalculated because the third user then becomes the second one and we could make use of s2,2 (which may be greater than s ~ , ~ ) in the new matrix.

Now we compare the complexity. It is evident that the ODFD without additional transform matrix is sim- pler than the DDFD. It is also noted that the match filter bank in the DDFD matches the signature sequences taking the values from only +1 and -1. Thus no multiplication is needed as the multiplication of -1 can be implemented by sign inversion. However, in the ODFD, the match filter bank matches the orthonormal functions which may take any real values, so that mul- tiplication is necessary. Fortunately, only one multipli- cation in the ODFD is needed for each match filter. An addition unit is required in each match filter for both DDFD and ODFD. Table 1 compares the complexity of the DDFD and ODFD. The number of multiplica- tions in the DDFD is about 1.5 times more than that of the ODFD. The difference becomes significant if the number of users increases. If K = 31, for example, the ODFD needs 961 multiplications while the DDFD needs 1426. The number of additions of the DDFD is twice as many as that of the ODFD.

Table 1: Complexity for a K-user DDFD and the ODFD systems

DDFD

Number of multiplications Number of additions K*

K(3 K-I )/2

Matrix inversion operation yes Matrix algorithm complexity exponential

Recalculation of matrix Yes Performance comparison with very close optimum

Near-far resistance near perfect

ODFD

KZ

K( K+ 1 )/2

no

linear

no

very close

near perfect

4 Results and discussion

The performance with a Gold code of degree 5 is given in Figs. 3-6. The BER against EblNo in the equal power case is shown in Figs. 3 and 4, where the two schemes (ODFD and DDFD) give very similar BER for both NF = 1 and NF = 2. The unequal power case is also considered and the results are shown in Figs. 5

and 6, where the power of the first user in the DDFD or the last user in the ODFD is 3dB less than that of the rest. The single user bound under this power level (3dB less than that of the rest) is also plotted. As expected, the ODFD performs equally well if compared with the DDFD. When NF = 2, performance for both the schemes is very close to the single user bound for either equal or unequal power case.

01

0.01

0.001 U m

0.0001

0 00001 t-I 2 3 L 5 6 7 a 9

BER for equal power 31-user system using Gold code of degree 5 Eb/N, , d B

Fig.3 IN, = I i (i) single'user bound (ii) DDFD, NF = 1, user 1 (iii) DDFD, NF = 1, user 31 (iv) ODFD, NF = 1, user 1 (v) ODFD, NF = 1, user 31

0.01

0 001 II: w m

0 0001

0.00001 2 3 li 5 6 I a 9

BER for equalpower 31-user system using Gold code of degree 5 Eb / N o , d B

Fig.4 (Nt = 2) (i) single user bound (ii) IDDFD, NF = 2, user 1 (iii) IDDFD, NF = 2, user 31 (iv) ODFD, NF = 2, user 1 (v) ODFD, Nr = 2, user 31

(i)'sin'gle user bound (ii) DDFD, NF = 1, user 1 (iii) DDFD, NF 7: 1, user 31 (iv) ODFD, NF =: 1, user 1 (v) ODFD, NF 1, user 31 (vi) single user bound 3dB less

IEE Puoc.-Commun., Vol. 144, No. 4, August 1997

0 1

0.01

L1: W m

0 ,001

0.0001

I

I I I ,

(i)'si&le uger bound (ii) IDDFD, N r = 2, user 1 (iii) IDDFD. N , = 2. user 3 1 (iv) ODFD, N ~ . = 2, user 1 (v) ODFD, NF = 2 , user 31 (vi) single user bound 3dB less

01

0.01

0.001 m

0 0001 1 * 8 (VI1

000001 L 1 I I , 2 3 L 5 6 7 8 9

Eb/No,dB Comparison of performance for d#eerent codes of degree 6 on Fig.7

equal power 6-user system using NF = I for ODFD (i) single user bound (ii) small set of Kasami (iii) M-sequence (iv) NO (v) GMW (vi) Gold (vii) large set of Kasami

To compare the performance of different codes, the number of users is kept to six. Fig. 7 compares the last user's performance for different codes (the degree of codes is 6) for NE. = 1 in the equal power case. The Gold code and large set of Kasami code offer the best performance. The small set of Kasami code and NO code also have the same performance. The GMW code performs worst, because its cross-correlation levels are relatively high if compared with those of the other codes. From our analysis, a 60-user Gold code system outperforms a six-user system using other codes, such as the small set of Kasami code, NO code and GMW code. Besides, the Gold code and large set of Kasami code maintain much larger code families than that of the other codes, making them ideal for the CDMA applications.

Fig. 8 shows the performance of different codes when two-feedback-path (NF = 2) detection is used. All codes perform better than given in Fig. 7 and they reach the single user bound except for the GMW code which may require even more feedback paths to enhance its performance.

The near-far resistance for the ODFD and DDFD is compared in Fig. 9 for a four-user system with both NF = 1 and NF = 2. The degree-3 Gold codes are con-

IEE Proc.-Commun., Vol. 144, No. 4, August 1997

sidered. It is shown that they have almost the same per- formance. If NF = 2, the BER of the first user for the ODFD and the last user for the DDFD is very close to the single user bound, indicating a near-perfect near- far resistance.

0.0001 1 , I , , I

2 3 L 5 6 7 8 Eb/N,,dB

Comparison of performance for dgerent codes of degree 6 on Fig.8 equal power 6-user system using NF = 2for ODFD (i) single user bound (ii) small set of Kasami (iii) M-sequence iiv) NO ("~GMW (vi) Gold (vii) large set of Kasami

0 0 1 , + (iil *. [viiil * iiv) * (vi1

II: W m

0.000001 I I I

0 1 2 3 L 5 6

BER offirst and last users in four-user system using Gold code of SNR(i l -SNR(L ),dB(i:I,2,31

Fig.9 degree 3 (i) single user bound, user 1 (ii) DBFD, N~ = 1, user 1 (iii) DDFD, NF = 2 , user 1 (ivi ODFD. N , = 1. user 1 ("j ODFD, = 2,'user 4 (vi) ODFD, NF = 1, user 4 (vii) ODFD, Nr; = 2, user 1 (viii) DDFD, NF = 1, user 4 (ix) ODFD, NF = 2, user 4

5 Conclusion

Based on the Gram-Schmidt orthogonalisation proce- dure, an orthonormal match filter bank can be applied to a synchronous CDMA multiuser detector, resulting in a novel ODFD multiuser detector. The performance of the ODFD is almost the same as that of the DDFD but with a much lower complexity because the trans- form matrix required in the DDFD is not needed. The ODFD algorithm needs no frequent updates in its orthonormal matrix even when the system parameters are changing. It can also use multiple feedback deci- sions to further improve the performance. The per- formance of the ODFD using different spreading codes, such as the Gold code, the large set of Kasami code etc., has been studied. It is shown that the Gold code and large set of Kasami code are promising candi- dates for use in the ODFD algorithm.

279

6 Acknowledgment 3 VARANASI, M.K., and AAZHANG, B 1 ‘Multistage detection in asynchronous code-division multiple-access communications’, IEEE Trans Commun , 1990, 38, pp. 509-519

The authors wish to thank the editor and the anony- 4 VARANASI, M K,, and AAZHANG, B ‘Near-optimm detec- mous reviewers for their comments and suggestions tion in synchronous code-division multiple-access systems’, IEEE

Trans Commun , 1991, 39, pp 125-136 which helped us greatly in revising the paper. 5 PATEL. P., and HOLTZMAN, J ‘Analysis of a simple succes-

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